We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one‐dimensional subspace by positive sparse forms involving Lp‐averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the Lp‐boundedness proved by Muscalu, Tao and Thiele, and entails as a corollary a novel rich multilinear weighted theory. A particular case of this theory is the Lqfalse(v1false)×Lqfalse(v2false)‐boundedness of the bilinear Hilbert transform when the weight vj belong to the class Aq+12∩RH2. Our proof relies on a stopping time construction based on newly developed localized outer‐Lp embedding theorems for the wave packet transform. In the Appendix, we show how our domination principle can be applied to recover the vector‐valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.

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A. We prove that bilinear forms associated to the rough homogeneous singular integralswhere Ω ∈ L q (S d −1 ) has vanishing average and 1 < q ≤ ∞, and to Bochner-Riesz means at the critical index in R d are dominated by sparse forms involving (1, p) averages. This domination is stronger than the weak-L 1 estimates for T Ω and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative A p -weighted estimates for Bochner-Riesz means and for homogeneous singular integrals with unbounded angular part, extending previous results of Hytönen-Roncal-Tapiola for T Ω . Our results follow from a new abstract sparse domination principle which does not rely on weak endpoint estimates for maximal truncations.

Abstract. We consider a Navier-Stokes-Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. The dissipative character of our model is weaker than the one where hereditary and instantaneous viscosity coexist, studied in [12] by Gal and Tachim-Medjo. Nevertheless, we prove the existence of a regular exponential attractor of finite fractal dimension under rather sharp assumptions on the memory kernel.

We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded L pextension to triples of intermediate UMD spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of UMD spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the UMD-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the UMD-valued bound for bilinear Calderón-Zygmund operators recently obtained by the same authors.